APPROXIMATE CONTROLLABILITY OF SECOND GRADE FLUIDS

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APPROXIMATE CONTROLLABILITY OF SECOND GRADE FLUIDS

Van-Sang Ngo, Geneviève Raugel

To cite this version:

Van-Sang Ngo, Geneviève Raugel. APPROXIMATE CONTROLLABILITY OF SECOND GRADE FLUIDS. 2019. �hal-01118496v2�

VAN-SANG NGO AND GENEVI `EVE RAUGEL

Abstract. This paper deals with the controllability of the second grade fluids, a class of non-Newtonian of differentiel type, on a two-dimensional torus. Using the method of Agrachev- Sarychev [1], [2] and of Sirikyan [26], we prove that the system of second grade fluids is approx- imately controllable by a finite-dimensional control force.

1. Introduction

The goal of this paper is to study the approximate controllability of the system of fluids of second grade, using low-mode (finite-dimensional) control forces. More precisely, we consider the following system

(1.1)







∂_t(u−α∆u)−ν∆u+ rot (u−α∆u)×u+∇p=f +η divu= 0

u(0) =u0,

on the domainT², which is the two-dimensional torus ]0,2πq₁[×]0,2πq₂[, withq₁>0 andq₂ >0.

Here u= (u₁(t, x), u₂(t, x)) andp=p(t, x) are unknown and represent the velocity vector field and the pressure function; f =f(t, x) is the extenal force field; and the control forceη=η(t, x) is supposed to belong to a finite-dimensional space which will be made more precise later.

Fluids of second grade belong to a particular class of non-Newtonian Rivlin-Ericksen fluids of differential type [25], which usually arise in petroleum industry, in polymer technology or in liquid crystal suspension problems. For these fluids, the Cauchy stress tensor σ is not linearly proportional to the local strain rate but given by

(1.2) σ =−pI+ 2νA₁+α₁A₂+α₂A²₁,

where ν stands for the kinematic viscosity,p is the pressure andA₁,A₂ represent the first two Rivlin-Ericksen tensors, which are

A₁(u) = 1

2 ∇u+∇u^T , corresponding to the local strain tensor and

A₂(u) = DA₁

Dt + (∇u)^T A₁+A₁(∇u), where

D

Dt =∂_t+u· ∇

is the material derivative. In [7], Dunn and Fosdick used the compatibility of (1.2) with ther- modynamics to prove that

α1+α2 = 0; α1 ≥0.

1991Mathematics Subject Classification. 35Q35, 93B05, 93C20.

Key words and phrases. Second grade fluid equations, approximate controllability, Agrachev-Sarychev method.

1

Setting α=α₁ and writing the equation Du

Dt =∂tu+u· ∇u= divσ+f, one obtain the equations of second grade fluids of the following form (1.3)







∂t(u−α∆u)−ν∆u+ rot (u−α∆u)×u+∇p=f inR+×T²

divu= 0 inR+×T²

u(0) =u₀ inT².

The local existence in time and uniqueness of a strong solution to (1.3) have been proven by Cioranescu and Ouazar in [6] in the case of two-dimensional or three-dimensional domains with non-slip boundary conditions. Moreover, the solution is global in time in the two-dimensional case. Second grade fluids in these domains were also studied by Moise, Rosa and Wang in [21], where the authors proved the existence of a compact global attractor in the two-dimensional case. The existence, the uniqueness of a strong solution and the dynamics of second grade fluids in the torus T² was studied in [24] by Paicu, Raugel and Rekalo, and in [23] by the first two authors, using the Lagrangian approach. For further results concerning the system (1.3), we refer the readers to [3], [4], [5], [13], [14], [15], [16], [17], [18], [20], . . .

In this paper, in order to study the approximate controllability of the second grade fluid system (1.1) by a low-mode controlη, we use the method introduced by Agrachev and Sarychev in [1] and [2] for the Navier-Stokes and Euler systems in the two-dimensional torus T². This method was extended later for the three-dimensional Navier-Stokes system by Shirikyan in [26]

and [27] and for the three-dimensional Euler system by Nersisyan in [22]. The main idea consists in proving that, if the (finite-dimensional) space of controlsE contains sufficiently many Fourier modes then, for any T > 0, the system (1.1) is approximately controllable in time T by an E-valued control η.

Before stating the main results and the main ideas of this paper, we will introduce the needed notations and function spaces. Let H^m(T²)² be the classical Sobolev space of two-dimensional vector fields, whose components belong to H^m(T²). For m = 0, we simply have H⁰(T²)² = L²(T²)². As in [24], for any m∈N, we denote V^m(T²)² the closure of the space

u∈C^∞(T²)²|uis periodic ,divu= 0, Z

T²

udx= 0

in H^m(T²)². Then V^m(T²)² is a Banach space, endowed with the classical norm of H^m(T²)². For any θ > 0, we define V^θ(T²)² using the method of interpolation between V^m(T²)² spaces.

Finally, we also use H_per^m (T²)² to denote the space of vector fields u ∈ H^m(T²)², which are periodic and whose mean value is zero.

In what follows, we recall the definition of a strong solution of the system (1.3).

Definition 1.1. LetT >0. For anyf ∈L^∞ 0, T, H_per¹ (T²)²

andu0 ∈V³(T²)², the vector field u(t, x) is said to be a strong solution of the system (1.3), with data(f, u0), on the time interval [0, T] if u ∈ C(0, T, V³(T²)²), ∂_tu ∈L^∞(0, T, V²(T²)²), u(0) = u₀, and for any t ∈]0, T], for any φ∈V⁰(T²)², the following equation holds

(1.4) h∂_t(u(t)−α∆u(t))−ν∆u(t) +rot(u(t)−α∆u(t))×u(t), φi=hf(t), φi. In [24], the authors prove that

Theorem 1.2. Let α >0 andT >0.

(1) For any f ∈ L^∞(0, T, H_per¹ (T²)⁾ and any u₀ ∈ V³(T²)², there exists a unique strong solution

u∈C(0, T, V³(T²)²)∩W^1,∞(0, T, V²(T²)²)

of the system (1.3). Moreover, for any t∈[0, T], the map V³(T²)² 3u₀ 7→u(t)∈V³(T²)² is continuous.

(2) Let m ≥ 1. Assume that f ∈ L^∞(0, T, H_per^m+1(T²)²) and u0 ∈ V^m+3(T²)². Then, the solution u of the system (1.3) belongs toC(0, T, V^m+3(T²)²).

For the system (1.1), we want to define the approximate controllability using low-mode con- trols. We will adapt the definition of approximate controllability given in [26] to the case of fluids of second grade.

Definition 1.3. Let θ >0, T >0 and let E be a finite-dimensional subspace of V³(T²)². The second grade fluid system (1.1) is θ-approximately controllable (θ-AC) in time T by E-valued controls if, for any ε >0, for any u₀, u_T ∈V³(T²), there exist a control η ∈L^∞(0, T, E)² and a strong solution u∈C(0, T, V³(T²)²) of the system (1.1) such that

ku(T)−uTk_Vθ(T²)² ≤ε.

For any m∈Z²\ {0}, let

c_m(x) =m^q,⊥coshm, xi_q and s_m(x) =m^q,⊥sinhm, xi_q,

wherem^q,⊥will be defined in Section 5. Then, it is classical that cm,sm, withm∈Z²\ {0}, are eigenvectors of the Stokes operator −P∆, where Pis the Leray projection, and that the family cm, sm|m∈Z²\ {0} forms an orthonormal basis ofV^k(T²q)²,k∈N. For anyN ∈N^∗, we set (1.5) H^N_q =Span{c_m, s_m |m∈Z\ {0},|m| ≤N}.

The main result of this paper is the following theorem.

Theorem 1.4. Let T > 0, f ∈ L^∞(0, T, H_per² (T²)²) and u0, uT ∈ V⁴(T²). Then the system (1.1) is3-AC in timeT by H³_q-valued controls.

We note that, unlike the case of Navier-Stokes equations, the system of second grade fluids is an exemple of asymptotically smooth system, which only possesses a smoothing effect in infinite time. The systems (1.1) or (1.3) also differ from theα-type models, such as the so-called α-Navier-Stokes system (see [8], [9] and the references therein). Indeed, the α-Navier-Stokes system contains the very regularizing term −ν∆(u−α∆u) instead of −ν∆u, and thus is a semilinear problem, which is easier to solve. It is different in the case of second grade fluids where the dissipation is much weaker. This weak smoothing effect explains why in our result, we can not obtain an approximate control in the same norm as the initial data but only a control in the weaker norm. We also remark the similar phenomenon in [24], where the Navier-Stokes system is proved to be the limit of the second grade fluid system inV^θ(T²)² for data inV³(T²)² (respectively in V³(T²)² for data inV⁴(T²)²).

Another problem when we want to apply the method of Shirikyan [26] lies in the complexity of the nonlinear term and the appearance of the term ∂_t(−α∆u). To avoid this difficulty, let

U =u−α∆u and U₀ =u0−α∆u0

and let us rewrite the system (1.1) in the following form (1.6)







∂tU +LU+B(U,U) =Pf+η divU = 0

U(0) =U₀, where

(1.7)

(LU =−νP∆(I−α∆)⁻¹U

B(U₁,U₂) =P rotU₁× (I−α∆)⁻¹U₂ .

Along with the system (1.6), we consider the following controlled system (1.8)







∂_tU +L(U+ζ) +B(U+ζ,U +ζ) =Pf+η divU = 0

U(0) =U₀.

As in [1] or [26], we give the following definition

Definition 1.5. For any finite-dimensional subspace E of V³(T²)², we define F(E) as the largest vector subspace of V³(T²)² such that, F(E)⊃E, and ifη ∈ F(E)\E then, there exist

k∈N^∗, α₁, . . . , α_k>0, η, ρ¹, . . . , ρ^k∈E satisfying

η=η−

k

X

j=1

αjB(ρ^j, ρ^j).

The main idea to prove Theorem 1.4 is to extend the space of control E to the larger space F(E). To this end, we will prove the following theorems.

Theorem 1.6. Let T > 0 and E a finite-dimensional subspace of V³(T²)². Then, the system (1.6) is (θ-) approximately controllable in time T by an E-valued control η if and only if so is the system (1.8) withE-valued controls η and ζ.

Theorem 1.7. Let T > 0 and E a finite-dimensional subspace of V³(T²)². Then, the system (1.8) is approximately controllable in time T by E-valued controls η and ζ if and only if so is the system (1.8) withF(E)-valued controls η.

As a consequence of these theorems, the approximate controllability by E-valued controls is equivalent to the approximate controllability byF(E)-valued controls. We can define a sequence of subspace

E=E₀⊂E₁⊂. . .⊂E_n⊂. . .

such that, for any n∈N we haveE_n+1 =F(E_n). The only problem is thatF(E) may be not larger than E. However, if we can chooseE in such a way that the space

E∞=

∞

[

n=0

En

is dense inV¹(T²)², then the approximate controllabilityE-valued controls will follows. Indeed, forT >0,ε >0,u₀, u_T ∈V⁴(T²)², we set

U₀ =u₀−α∆u₀ and U_T =u_T −α∆u_T. Then, we can always exactly control the system (1.6) by the control

η=∂tU+LU+B(U)−Pf ∈L^∞(0, T, V¹(T²)²), where

U(t) = 1

T (I−α∆) ((T −t)u₀+tu_T).

Thus, if E∞ is dense in V¹(T²)², then there exists n ∈ N large enough such that the system (1.6) is approximately controllable byEn-valued controls. Using Theorems 1.6 and 1.7, we can prove by induction that (1.6) is approximately controllable by E-valued controls. In order to prove Theorem 1.4, we only need to prove the following result

Theorem 1.8. If E =H³_q thenE∞⊃ H_q^N, for anyN ∈N, N ≥3.

The paper will be organized as follows. In Section 2, we study a pertubation of the system (1.6), which is necessary to prove our main theorem. Theorem 1.6 will be proved in Section 3.

Section 4 is devoted to the demonstration of Theorem 1.7. In Section 5, we put in evidence the saturation property given in Theorem 1.8. Finally, in the last section, we wil prove the main theorem 1.4.

2. Preliminary results on the system of fluids of second grade In this section, we consider the following perturbed system of fluids of second grade

(2.1)







∂_tW+LW+B(W) +B(W,V) +B(V,W) =Pf inT²×[0, T]

divW = 0 inT²×[0, T]

W(0) =W₀ inT²,

where V ∈L^∞(0, T, V²(T²)²),f ∈L^∞(0, T, H_per¹ (T²)²). We want to prove that Theorem 2.1. Let T >0 fixed.

1. For any V ∈ L^∞(0, T, V²(T²)²), f ∈ L^∞(0, T, H_per¹ (T²)²) and W₀ ∈ V¹(T²)², the system (2.1) has a unique solution

W ∈L^∞(0, T, V¹(T²)²)∩C(0, T, V¹(T²)²).

Moreover, if V ∈L^∞(0, T, V³(T²)²),f ∈L^∞(0, T, H_per² (T²)²) andW₀∈V²(T²)² then W ∈L^∞(0, T, V²(T²)²)∩C(0, T, V²(T²)²).

2. Let W and Wc be two solutions of the system (2.1), corresponding to data (V, f,W₀) and (V,b f ,bWc₀) respectively. Then, if

W ∈c L^∞(0, T, V²(T²)²)∩C(0, T, V²(T²)²), then, there exists a constant C >0 such that, for any t∈[0, T], we have (2.2)

W(t)−W(t)c V¹(T²)²

≤C

V −Vb

L²(0,T ,V²)² +

Pf −Pfb

L²(0,T ,V¹)² +

W₀−Wc₀ V¹(T²)²

.

We remark that if we set







v= (I−α∆)⁻¹V w0 = (I−α∆)⁻¹W₀

w= (I−α∆)⁻¹W then, w is solution of the following system

(2.3)







∂_t(w−α∆w)−ν∆w+PB(w) +PB(w, v) +PB(v, w) =Pf divw= 0

w(0) =w0, where

B(u1, u2) = rot (u1−α∆u1)×u2 and B(u) =B(u, u).

Theorem 2.1 is in fact equivalent to the following theorem for the system (2.3)

Theorem 2.2. Let T >0 fixed.

1. For any v∈L^∞(0, T, V⁴(T²)²),f ∈L^∞(0, T, H_per¹ (T²)²) andw0 ∈V³(T²)², the system (2.3) has a unique solution

w∈L^∞(0, T, V³(T²)²)∩C(0, T, V³(T²)²).

Moreover, if v∈L^∞(0, T, V³(T²)²), f ∈L^∞(0, T, H_per² (T²)²) and w₀ ∈V²(T²)² then w∈L^∞(0, T, V⁴(T²)²)∩C(0, T, V⁴(T²)²).

2. Let w and wb be two solutions of the system (2.3), corresponding to data (v, f, w0) and (bv,f ,bwb0) respectively. Then, if

wb∈L^∞(0, T, V⁴(T²)²)∩C(0, T, V⁴(T²)²), then, there exists a constant C >0 such that, for any t∈[0, T], we have (2.4) kw(t)−w(t)kb _V3(T²)²

≤C

kv−vkb _L2(0,T ,V⁴)² +

Pf −Pfb

L²(0,T ,V¹)² +kw₀−wb0k_V3(T²)²

.

We remark that the proof of the existence of solutions of the systems (2.1) and (2.3) follows the lines of the proof of [[24], Theorems 2.1 and 2.4]. In what follows, we give the needed a priori estimates to prove (2.2) and (2.4). We will set

(2.5)

W = rot (w−α∆w), Wc= rot (wb−α∆w)b V = rot (v−α∆v), Vb= rot (bv−α∆bv) O=W −W.c

2.1. Propagation of theV³-norm. In this paragraph, we give a priori estimates of a solution of the system (2.3) in V³(T²)-norm. Applying the rot operator to the first equation of (2.3), we obtain

(2.6) ∂tW+ ν

αW+P(w· ∇W) +P(v· ∇W) +P(w· ∇V) = rotPf+ ν αrotw.

Since v and ware divergence-free vector fields on T², integrations by parts show that hw· ∇W,Wi_L2(T²)² =hv· ∇W,Wi_L2(T²)² = 0.

As a consequence, taking the L²(T²)² inner product of (2.6) with W, we get 1

2 d

dtkWk²_L2(T²)² +ν

αkWk²_L2(T²)²

(2.7)

≤

hw· ∇V,Wi_L2(T²)²

+

hrotf,Wi_L2(T²)²

+ν

α

hrotw,Wi_L2(T²)²

≤ krotfk_L2(T²)²kWk_L2(T²)² +

k∇Vk_L2(T²)² +ν α

kWk²_L2(T²)², which implies that

(2.8) d

dtkWk²_L2(T²)² ≤ kfk²_V1(T²)² + 2

1 +kvk_V4(T²)²

kWk²_L2(T²)². Finally, the Gronwall lemma gives, for any 0≤t≤T,

(2.9) kW(t)k²_L2(T²)² ≤

kW(0)k²_L2(T²)²+kfk²_L∞(0,T ,V¹(T²)²)

exp

n 2t

1 +kvk_L∞(0,T ,V⁴(T²)²)

o .

2.2. A priori estimates of the difference of two solutions in V³-norm. Using the same notations as in [6], [24] or [23], we identify a 2D vector (u1, u2) with the 3D vector (u1, u2,0) and a scalarλwith the 3D vector (0,0, λ). We also recall the identity

rot (rot (a)×b) =b· ∇rot (a),

where a and b are 2D vector fields and divv = 0. We deduce from (2.3) that W and Wc are solutions of the following equation, with data (v, f, w₀) and (bv,f ,bwb₀) respectively.

(2.10) ∂_tW+ ν

αW+P(w· ∇W) +P(v· ∇W) +P(w· ∇V) = rotPf+ ν αrotw.

The calculation of the difference between the equations corresponding to W and Wcshows that O=W −Wcsatisfies the following equation

(2.11) ∂tO+ ν

αO+P(w· ∇O) +P

(w−w)b · ∇Wc

+P(v· ∇O) +P

(v−bv)· ∇Wc +P

w· ∇

V −Vb +P

(w−w)b · ∇Vb

= rot (Pf −Pfb) + ν

αrot (w−w).b Next, we will take theL² inner product of (2.11) with O.

Using the divergence-free property ofv and w, we have

(2.12) hv· ∇O,Oi_L2(T²)² =hw· ∇O,Oi_L2(T²)² = 0.

Now, using H¨older’s and Cauchy-Schwarz’s inequalities, we get

D

(w−w)b · ∇cW,OE

L²(T²)²

≤ ∇Wc

L²(T²)²

kw−wkb _L∞(T²)²kOk_L2(T²)²

(2.13)

≤C ∇cW

L²(T²)²

kw−wkb _V3(T²)²kOk_L2(T²)²

≤Ckwkb _V4(T²)²kOk²_L2(T²)². The same calculations give

D

(v−bv)· ∇Wc,OE

L²(T²)²

≤C ∇Wc

_L2(

T²)²kv−bvk_L∞(T²)²kOk_L2(T²)²

(2.14)

≤Ckwkb _V4(T²)²

kv−bvk²_V3(T²)²+kOk²_L2(T²)²

,

D

w· ∇

V −Vb ,OE

L²(T²)²

(2.15)

≤ kwk_L∞(T²)²

∇

V −Vb L²(T²)²

kOk_L2(T²)²

≤Ckwk_L∞(T²)²kv−bvk_V4(T²)²kOk_L2(T²)²

≤C

kw−wkb _L∞(T²)² +kwkb _L∞(T²)²

kv−vkb _V4(T²)²kOk_L2(T²)²

≤C

kOk_L2(T²)² +kwkb _L∞(T²)²

kv−bvk_V4(T²)²kOk_L2(T²)²

≤Ckwkb _L∞(T²)²kv−bvk²_V4(T²)² +C

kwkb _L∞(T²)² +kv−bvk_V4(T²)²

kOk²_L2(T²)²,

D

(w−w)b · ∇bV,OE

L²(T²)²

≤ kw−wkb _L∞(T²)²

∇bV

L²(T²)²kOk_L2(T²)²

(2.16)

≤Ckw−wkb _V3(T²)²kbvk_V4(T²)²kOk_L2(T²)²

≤Ckbvk_V4(T²)²kOk²_L2(T²)².

For the forcing term, we have (2.17)

D

rot (Pf−Pfb),OE

L²(T²)²

≤C

Pf −Pfb

2

V¹(T²)² +kOk²_L2(T²)²

, and finally,

(2.18)

hrot (w−w),b Oi_L2(T²)²

≤ krot (w−w)kb _L2(T²)²kOk_L2(T²)² ≤CkOk²_L2(T²)². Now, using Estimates (2.12) to (2.18), we obtain

1 2

d

dtkOk²_L2(T²)²+ ν

αkOk²_L2(T²)²

≤Ckwkb _V4(T²)²kv−vkb ²_V4(T²)²+C

Pf −Pfb

2 V¹(T²)²

+C

1 +kwkb _V4(T²)²+kbvk_V4(T²)² +kv−bvk_V4(T²)²

kOk²_L2(T²)². For any 0≤t≤T, the Gronwall lemma implies that

(2.19) kO(t)k²_L2(T²)² ≤Ce^C2t

kw(0)−w(0)kb ²_V3(T²)²

+T

Pf−Pfb

2

L^∞_t V¹(T²)² +C1Tkv−bvk²_L∞ t V⁴(T²)²

, where C is a generic positive constant and

C₁ =kwkb _L∞(0,T ,V⁴(T²)²),

C₂ = 1 +kwkb _L∞(0,T ,V⁴(T²)²)+kbvk_L∞(0,T ,V⁴(T²)²)+kv−bvk_L∞(0,T ,V⁴(T²)²). The inequality (2.4) of Theorem 2.2 is then proved.

3. Study of the extended controlled system

In this section, we want to show that the approximate controllability of the system (1.6) is equivalent to the approximate controllability of the system (1.8) by low-mode controls. For any finite-dimensional subspaceE ofV³(T²)², we remark that the approximate controllability of the system (1.6) by E-valued controls implies immediately the approximate controllability of the system (1.8) in the same space of controls. Indeed, we only need to chooseζ = 0 in the system (1.8). Then, in order to prove Theorem 1.6, we only need to prove the following result

Theorem 3.1. Let T > 0 and E be a finite-dimensional subspace of V³(T²)². Let η, ζ in L^∞(0, T, E)² and

U ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)²

be a solution of (1.8). Then, for any k ∈ N^∗, there are a control η_k ∈ L^∞(0, T, E)² and a solution

U_k ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)² of the system (1.6), with η=η_k, such thatU_k(0) =U₀, and

kU_k(T)− U(T)k_V3(T²)² ≤ 1 k.

Proof. First of all, we can rewrite the system (1.8) as (3.1)







∂tU+LU+B(U,U) +B(U, ζ) +B(ζ,U) =Pf+η− Lζ− B(ζ, ζ) divU = 0

U(0) =U₀,

where

LU=−νP∆(I −α∆)⁻¹U and B(U₁,U₂) =P rotU₁× (I−α∆)⁻¹U₂ .

Thus, applying Theorem 2.1 to this system, for anyη, ζ ∈L^∞(0, T, E)², we obtain the existence of a unique solution of the system (3.1) (or (1.8))

U ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)² . Next, we remark that we can also rewrite the system (1.8) as

(3.2)







∂t(U +ζ) +L(U +ζ) +B(U +ζ) =Pf+ ˜η divU = 0

U(0) =U₀, where

˜

η =η+∂_tζ,

which means that, if U is a solution of the system (1.8) and if ˜η belongs toL^∞(0, T, E)², then U+ζ is a solution of the system (1.6), withη replaced by ˜ηand U₀ byU₀+ζ(0). So, if we want to construct a solution of the controlled system (1.6) satisfying the conditions of Theorem 3.1, we only need to check whether the conditions at timet= 0 andt=T are satisfied. To this end, we will consider a sequence of controls

ζl∈C¹(0, T, E), ∀l∈N^∗ such that

ζ_l(0) =ζ_l(T) = 0 and

l→+∞lim kζ_l−ζk_L2(0,T ,V²(T²)²)= 0.

Applying Theorem 2.1 to the system (3.1), for any l∈N^∗, there exists a unique solution U_l∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)²

of the system (3.1) (or (1.8)), with ζ replaced by ζl. Moreover, for any k ∈ N^∗, there exists l0 ∈N^∗ such that, for any l≥l0, for anyt∈[0, T], we have

U_l(t)− U(t)

V¹ ≤C

kζ_l−ζk_L2(V²)+kLζ_l− Lζk_L2(V¹)+kB(ζ_l, ζ_l)− B(ζ, ζ)k_L2(V¹)

≤ 1 k. Now, we set

U_k =U_l0 +ζ_l0 and η_k=η+∂_tζ_l0.

Then, U_k is the solution of the system (1.6), with η replaced by ηk ∈L^∞(0, T, E)². Moreover, we have

U_k(0) =U_l0(0) +ζ_l0(0) =U_l0(0) =U₀, and

U_k(T)− U(T)

V¹(T²)² =

U_l0(T)− U(T)

V¹(T²)² ≤ 1 k. Theorem 3.1 is proved.

4. Convexification of the controlled system

This section is devoted the to the proof of Theorem 1.7. From the definition 1.5 ofF(E), we remark that E ⊂ F(E) and so, the approximate controllability of the system (1.1) (or (1.6)) by E-valued controls evidently implies the approximate controllability of the system (1.1) (or (1.6)) byF(E)-valued controls. In order to prove Theorem 1.7, we only need to prove that Theorem 4.1. Let T >0, E be a finite-dimensional subspace of V³(T²)². Let η be a control in L^∞(0, T,F(E))² and

U ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)²

be a solution of (1.6), with η replaced by η. Then, for any k ∈N^∗, there are controls η_k, ζ_k ∈ L^∞(0, T, E)² and a solution

U_k ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)²

of the system (1.8), with η replaced by η_k and ζ by ζ_k, such that U_k(0) =U₀, and

k→+∞lim

U_k(T)− U(T)

V¹(T²)² = 0.

The proof of Theorem 4.1 will be divided into several steps.

Step 1. Reduction of the proof of Theorem 4.1 to the case of F(E)-valued piecewise constant in time controls.

We suppose that Theorem 4.1 is true for F(E)-valued piecewise constant controls. We want to prove that Theorem 4.1 is also true in the general case. Letη ∈L^∞(0, T,F(E)) and let

U ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)²

be a solution of (1.6), with η replaced byη. We consider an approximation of η by a sequence {η^m} ofF(E)-valued piecewise constant in time controls such that

m→+∞lim kη^m−ηk_L2(0,T ,V²(T²)²)= 0.

Applying Theorem 2.1 (while taking V = 0 and replacing f by f +η^m and f +η), we deduce the existence of a solution

U^m ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)² of the system (1.6), withη replaced by η^m, such that

sup

t∈[0,T]

U^m(t)− U(t)

V¹(T²)² ≤C U

L²(0,T ,V²(T²)²)

kη^m−ηk_L2(0,T ,V²(T²)²)≤ ε 2, for any m larger than a certain m0 ∈N^∗.

If Theorem 4.1 is true for F(E)-valued piecewise constant controls, we deduce the existence of η and ζ inL^∞(0, T, E)², and a solution

U ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)² of the system (1.8) such that

U(0) =U^m0(0) =U(0), and

sup

t∈[0,T]

kU(t)− U^m0(t)k_V1(T²)² ≤ ε 2. Therefore,

sup

t∈[0,T]

U(t)− U(t)

V¹(T²)² ≤ sup

t∈[0,T]

kU(t)− U^m0(t)k_V1(T²)²+ sup

t∈[0,T]

U^m0(t)− U(t)

V¹(T²)² ≤ε.

Remark 4.2. Using an argument by iteration, we can reduce the study to the case where the control η is constant in time. So from now on, we will considerη ∈ F(E), which is constant in the time variable.

Step 2. Construction of solutions of the extended controlled system (1.8).

The construction of controlsη_k,ζ_kand a solutionU_kof the controlled system (1.8), with (η, ζ) replaced by (η_k, ζ_k), follows the lines of the construction in [26] (see also [1] and [2]). We recall that

U ∈L^∞ 0, T, V²(T²)²

∩C 0, T, V²(T²)²

is the solution of the controlled system (1.6), withη replaced byη ∈ F(E). Letε >0 andδ >0 which will be made more precise later and choose N >0 large enough such that

kPf−P_NPfk_L2(0,T ,V¹(T²)²)+kU₀−P_NU₀k_V1(T²)² ≤δ,

where P_N is the projection onto the space of the first N eigenvectors of the Stokes operator

−P∆ and where P is the Leray projection onto the subspace of divergence-free vector fields of L²(T²)². Let

V₀ =P_NU₀ and V_N be the solution of the system

(4.1)







∂tV_N+LV_N +B(V_N) =PPNf +η divV_N = 0

V_N(0) =V₀ =P_NU₀.

Using the definition in (1.7), we remark that (I−α∆)⁻¹V_N is the solution of the system (1.1), with u0 = (I −α∆)⁻¹PNU₀ and η replaced by η. Then, applying [[24], Theorems 2.1 and 2.4], we obtain the existence of a unique solution

V_N ∈L^∞ 0, T, V³(T²)²

∩C 0, T, V³(T²)²

of the system (4.1). The following lemma (see [26]) allows us to have a “good decomposition”

of F(E)-valued controls in terms ofE-valued controls.

Lemma 4.3. Let E be a finite-dimensional subspace ofV³(T²)². Then, for anyη ∈ F(E)\E, there exist m ∈ N^∗; η, ρ¹, . . . , ρ^m ∈ E and λ1, . . . , λm ∈ R^∗+, with Pm

j=1λj = 1, such that, for any U ∈V²(T²)², we have

B(U)−η=

m

X

j=1

λ_j B U+ρ^j

+Lρ^j

−η.

Proof. Since η∈ F(E)\E, Definition 1.5 implies that there exist k∈N^∗; α1, . . . , α_k >0; and ˜η,ρ˜¹, . . . ,ρ˜^k∈E such that

η = ˜η−

k

X

j=1

αjB( ˜ρ^j).

Let m= 2k,α=α₁+. . .+α_k and







λ_j = αj

2α, ρ^j =√

αρ˜^j, ∀j∈ {1, . . . , k}

λ_j = αj−k

2α , ρ^j =−√

αρ˜^j−k, ∀j∈ {k+ 1, . . . , m}. We remark that, for anyj ∈ {1, . . . , k}, we have

λj =λ_j+k and ρ^j =−ρ^j+k.

Then, for any U ∈V²(T²)², direct calculations give, B(U,U)−η =

m

X

j=1

λj B(U +ρ^j) +Lρ^j

−η.

Lemma 4.3 allows us to rewrite the system (4.1) as follows

(4.2)















∂tV_N +LV_N+

m

X

j=1

λj B(V_N +ρ^j) +Lρ^j

=PPNf+η divV_N = 0

V_N(0) =V₀ =P_NU₀.

Now, we use the same construction as explained in [26] to build the needed additional control ζ in the system (1.8). To this end, we introduce the following 1-periodic functionϕ : R+→E (4.3)







ϕ(s) =ϕ(s+ 1) for anys∈R+

ϕ(s) =ρ¹ si 0≤s < λ₁

ϕ(s) =ρ^j siλ1+. . .+λj−1 ≤s < λ1+. . .+λj, for any 2≤j ≤m For any k≥1, let

(4.4) ψ_k(t) =ϕ

kt T

. Then, the system (4.2) can be rewritten as follows (4.5)







∂_tV_N +L(V_N +ψ_k) +B(V_N+ψ_k) =PP_Nf+η+f_k divV_N = 0

V_N(0) =V₀ =PNU₀, where

(4.6) fk(t) =gk(t) +hk(t),

with

(4.7)















gk(t) =Lψ_k(t)−

m

X

j=1

λjLρ^j

hk(t) =B(V_N +ψk(t))−

m

X

j=1

λjB(V_N+ρ^j).

We remark that, for anys∈ {1,2} and for anyU ∈V^s(T²)², we have kLU k_Vs(T²)² ≤ ν

αkU k_Vs(T²)². Then, for s∈ {1,2} and for any t≥0, simple calculations give,

kg_k(t)k_Vs(T²)² ≤ kLψ_k(t)k_Vs(T²)² +

m

X

j=1

λ_j Lρ^j

V^s(T²)² ≤ 2ν α max

1≤j≤m

ρ^j V^s(T²)²

(4.8)

kh_k(t)k_Vs(T²)² ≤ kB(V_N +ψk(t))k_Vs(T²)² +

m

X

j=1

λj

B(V_N +ρ^j) V^s(T²)²

(4.9)

≤2 max

1≤j≤m

B(V_N +ρ^j)

V^s(T²)².

Concerning the bilinear operator B, classical results imply, for any U ∈ V³(T²)² and for any V ∈V²(T²)²,

kB(U,V)k_V1(T²)² ≤

rot rotU × (I −α∆)⁻¹V L²(T²)²

(4.10)

=

(I−α∆)⁻¹V

· ∇(rotU)

L²(T²)² ≤C(α)kVk_V2(T²)²kU k_V2(T²)²

and

kB(U,V)k_V2(T²)² ≤C(α)

krotU k_V2(T²)²

(I−α∆)⁻¹V L^∞(T²)²

(4.11)

+krotU k_L∞(T²)²

(I−α∆)⁻¹V V²(T²)²

≤C(α)kVk_V2(T²)²kU k_V3(T²)². Then, we obtain, for s∈ {1,2}and for any t≥0,

(4.12) kg_k(t)k_Vs(T²)² ≤ 2ν α max

1≤j≤m

ρ^j V^s(T²)²

and

(4.13) kh_k(t)k_Vs(T²)² ≤2C(α) max

1≤j≤m

V_N +ρ^j V²(T²)²

V_N +ρ^j

V^s+1(T²)².

Next, for any f ∈L^∞(0, T, H_per¹ (T²)²), we letKf be the solution of the system (4.14)







∂tZ+LZ =Pf divZ= 0 Z(0) = 0.

By considering z= (I−α∆)⁻¹Z and applying [[24], Theorem 2.4], we have the following Lemma 4.4. Let s∈N^∗. Iff ∈L^∞(0, T, V^s(T²)²) then

Kf ∈L^∞(0, T, V^s(T²)²)∩C(0, T, V^s(T²)²).

For any k∈N^∗, we set

W_k=V_N − Kf_k. Then, the system (4.5) becomes

(4.15)











∂_tW_k+LW_k+B(W_k) +B(W_k, ψ_k+Kf_k) +B(ψ_k+Kf_k,W_k)

=PP_Nf+η−PB(ψ_k+Kf_k)− Lψ_k divW_k= 0

W_k(0) =V_N(0) =PNU₀.

In other words, W_k is the solution of the system (2.1), with data (V, f,W₀) replaced by (ψ_k+Kf_k, P_Nf +η− B(ψ_k+Kf_k)− Lψ_k, P_NU₀).

Let

V =ψ_k, f =f+η− B(ψ_k)− Lψ_k, W₀ =U₀,

and letU_k be the solution of the system (2.1) with data (V, f ,W₀). It is then easy to show that U_k is the solution of the controlled system (1.8), with controlsη and ζ =ψ_k







∂tU_k+L(U_k+ψk) +B(U_k+ψk) =Pf +η divU_k= 0

U_k(0) =U₀.